A triangle possesses rotational symmetry only if it is an equilateral triangle.
Understanding Rotational Symmetry
Rotational symmetry occurs when a shape can be rotated around a central point by less than a full turn (360 degrees) and still appear exactly as it did in its original position. The number of times it matches its original appearance during a full rotation determines the order of rotational symmetry.
The Equilateral Triangle and Rotational Symmetry
Based on geometric principles, and as stated in our reference:
"The only type of triangle that has rotational symmetry is an equilateral triangle, or a triangle in which all of the sides have equal length and all angles have an equal measure of 60°".
This is the key rule regarding rotational symmetry in triangles. Other types of triangles, such as isosceles or scalene triangles, do not have this property (beyond the trivial 360-degree rotation which all shapes have).
Properties of an Equilateral Triangle
An equilateral triangle is uniquely suited for rotational symmetry because of its specific properties:
- All three sides are of equal length.
- All three internal angles are equal, each measuring exactly 60°.
These perfect equalities ensure that rotating the triangle by a specific angle results in a configuration indistinguishable from the start.
How Rotation Works for Equilateral Triangles
An equilateral triangle has rotational symmetry of order 3. This means it can be rotated around its center point three times (including the 360° mark) and look identical to its starting position each time. The angles of rotation that achieve this are:
- 120°: Rotating an equilateral triangle by 120° about its centroid (the intersection of its medians) brings one vertex to the position previously occupied by another, with the sides and angles aligning perfectly.
- 240°: A further 120° rotation (totaling 240°) again brings the third vertex into the position of the first, maintaining the original appearance.
- 360°: A full rotation returns the triangle to its absolute original position.
Other Triangle Types
To illustrate why only equilateral triangles have non-trivial rotational symmetry, consider other types:
| Triangle Type | Side Lengths | Angle Measures | Rotational Symmetry (Order) | Notes |
|---|---|---|---|---|
| Equilateral | All 3 equal | All 3 equal (60°) | 3 | Looks same after 120°, 240° rotations |
| Isosceles | 2 equal | 2 equal | 1 | Only looks same after 360° rotation |
| Scalene | All 3 different | All 3 different | 1 | Only looks same after 360° rotation |
As the table shows, only the equilateral triangle has an order of rotational symmetry greater than 1.
Key Takeaway
In summary, a triangle possesses rotational symmetry specifically when it is an equilateral triangle, due to its equal sides and equal angles allowing it to perfectly overlap itself after rotations of 120° and 240°.
